Computational Procedures for the Multi-Disciplinary ... - CiteSeerX

Computational Procedures for the Multi-Disciplinary Constrained Optimization of Wind Turbines C.L. Bottasso∗, F. Campagnolo, A. Croce Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano, Italy Scientific Report DIA-SR 10-02 January 2010 (Revised May 2011)

Abstract We describe procedures for the multi-disciplinary design optimization of wind turbines, where design parameters are optimized by maximizing a merit function, subjected to constraints that translate all relevant design requirements. Evaluation of merit function and constraints is performed by running simulations with a parametric high-fidelity aero-servo-elastic model; a detailed cross sectional structural model is used for the minimum weight constrained sizing of the rotor blade. To reduce the computational cost, the multi-disciplinary optimization is performed by a multi-stage process that first alternates between an aerodynamic shape optimization step and a structural blade optimization one, and then combines the two to yield the final optimum solution. A complete design loop can be performed using the proposed algorithm using standard desktop computing hardware in one-two days. The design procedures are implemented in a computer program and demonstrated on the optimization of multi-MW horizontal axis wind turbines and on the design of an aero-elastically scaled wind tunnel model.

Notation B c CP CF D E LDLC P Py R Sk T t V

Number of blades Blade chord Power coefficient Thrust coefficient List of given input data Load envelope List of Design Load Conditions Power Annual energy production Rotor radius Generic kth blade cross section Torque Time Wind speed

∗

Corresponding author, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano, Via La Masa 34, 20156 Italy. E-mail: [email protected]; Tel.: +39-02-2399-8315; Fax: +39-02-2399-8334.

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1 Introduction

vtip w d l n pa ps s σ β δtip η λ Ω ω θ (·)∗ (·)k (·)T (·)r (·)II (·)II 1/2 (·)III (·)M (·)P {·}

1

2

Blade tip speed Weight Vector of fatigue damage indices at all sectional points of interest Vector of externally applied loads Vector of shape functions Vector of aerodynamic design parameters Vector of structural design parameters Vector of sectional stress resultants Vector of maximum strains at all sections of interest Vector of maximum stresses at all sections of interest Blade pitch Blade tip deflection Non-dimensional blade span-wise coordinate η ∈ [0, 1] Tip-speed-ratio Rotor speed Structural natural frequency Blade twist Optimal quantity Quantity pertaining to the kth blade section Transpose Rated quantity, i.e. corresponding to the achievement of rated power Quantity pertaining to region II Quantity pertaining to region II1/2 Quantity pertaining to region III Quantity pertaining to the scaled model Quantity pertaining to the full scale system List of unordered data

Introduction

The optimization of the configuration of a wind turbine is a complex multi-disciplinary problem. Many considerations of various nature must be made and taken into account in order to arrive at a design that achieves the right trade-offs between performance and cost, while accounting for a variety of other constraints that make that specific design solution viable from all relevant points of view. Clearly, tools that can effectively support such complex design efforts in a integrated, holistic, manner and with rapid turn-around times can be useful for sizing a new machine, for improving a tentative configuration, or for studying modifications to existing models. The optimization of wind turbines has been the subject of a number of investigations in the recent literature. For example, Refs. [1, 2, 3, 4] describe procedures for the determination of the optimal aerodynamic shape of rotor blades. Typically, such approaches use aerodynamic wind turbine models based on variants of the blade element momentum theory. On the other hand, Ref. [5] adopts a more sophisticated aero-elastic model to account for the structural dynamics response of the machine, although even in this case the optimization is limited to the aerodynamic shape of the blade and does not account for the structural sizing aspect of the problem. The purely structural sizing problem given an aerodynamic shape has been studied in Ref. [6], and specialized FEM-based software for the detailed structural analysis of rotor blades is described in Ref. [7].

1 Introduction

3

On the other hand, the integrated multi-disciplinary optimization of wind turbine rotors addresses a much more complex problem, that considers the aerodynamic shape optimization, the evaluation of all relevant load conditions (which in turn requires the definition of appropriate control laws), the optimal sizing of the structural members under the effects of the loads, considering the mutual couplings between the various sub-disciplines and simultaneously accounting for the presence of a number of design constraints of various nature. It appears that there are very few holistic design tools with such characteristics, and few papers have been devoted to the subject (see Refs. [8, 9]). Most notably, the two codes RotorOpt [9, 10] and FOCUS5 [11] implement integrated design environments; on the other hand, Ref. [12] describes a suite of tools that cover all primary modeling needs required to perform a full design cycle, although they do not appear to have been yet cast within a unified optimization framework. In this work, we describe a suite of procedures for the integrated multi-disciplinary constrained optimization of wind turbines, which includes aerodynamics, load calculation, synthesis of control laws and structural sizing. Our proposed approach, although similar in spirit and motivation to the few other published holistic design tools, has two principal distinguishing features. First, we use models that can capture the relevant aero-servo-elastic characteristics of the system to a high level of fidelity. The use of sophisticated models of the machine since the very inception of the design, implies that one can immediately account for some important aspects (e.g., the placement of the rotor natural frequencies, the effects on structural blade sizing induced by constraining the maximum tip deflection and fatigue damage, and several others), which therefore can express their effects and couplings with all other design requirements; with simpler models some of these interactions are lost or not properly accounted for, so that the obtained design must be modified a posteriori to include the overlooked effects. The approach described here is based on a two-level modeling system. The first level model is a parametric global model of a wind turbine, implemented within a comprehensive aero-servo-elastic non-linear finiteelement-based multibody dynamics solver. The model enables the evaluation of a variety of contributors to the merit function and constraints, by running simulations that include eigenanalysis and transient design load cases according to Refs. [13, 14], as necessary. The second level model is a finite element parametric cross sectional model of the blade, that enables the calculation of section-wise stresses and strains under the loads computed on the first model, to support the determination of a minimum weight blade configuration satisfying all associated required constraints. The two-level modeling is closed by synthesizing on the detailed blade model beam-equivalent structural and inertial characteristics that are then used in the definition of the multibody model. Second, we formulate many of the complex considerations, which are made by the designer so as to ensure a viable solution, as constraints to the optimization problem. Although this has been in part done in previous works on the subject, in the present effort we have tried to include as many of the crucial design constraints as possible. In fact, many constraints have profound couplings with the functioning of the machine and hence have complex effects on its design. For example, the inclusion of a noise constraint even through a simple limit on the tip speed, alters the regulation strategy of the machine in between regions II (partial load) and III (full load) [15], which has an effect on the power curve and hence on the annual energy production. This effect is mitigated by changing the rotor solidity, that in turn alters the loads and might interact with other geometric constraints, for example on the maximum chord length for ensuring the transportability of the blade on-board trucks. Another constraints with complex effects on the design is the relative placement of the rotor natural frequencies with respect to the predominant harmonic excitations, which should be done correctly to avoid the insurgence of resonant conditions within the whole operating envelope of the machine. Clearly, this can only

2 Multi-disciplinary Optimization Algorithm

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be done while simultaneously ensuring a blade that, under the maximum experienced loads, does not exceed the maximum allowable stresses and strains, does not have excessive deflections and is also of minimum possible weight. For capturing the effects of such constraints in a correct way, they need to be consistently enforced during the design optimization, which is one of the highlights of the present approach. The paper is organized according to the following plan. At first, we describe the multidisciplinary optimization algorithm in Section 2. Next we describe the models used for supporting the various phases of the analysis and their implementation in Section 3. First we describe in §3.1 the aero-servo-elastic global multibody models of the wind turbine. Next, we give the parameterization of the aerodynamic model in §3.2, i.e. the choice of the aerodynamic design parameters. This is followed by the description of the structural design parameters and the cross sectional modeling in §3.3. In order to support the analyses described above within an optimization loop, it is necessary to completely automate the synthesis of a complex set of supporting features that are described in §3.4. Specifically, we describe the computation of the curves of the power coefficient versus tip speed ratio for varying blade pitch setting (§3.4.1), which enables the determination of the regulation policy across the whole operating range of wind speeds (§3.4.2). This in turn is necessary for the synthesis of a pitch-torque controller (§3.4.3), needed for conducting all transient simulations (§3.4.4) that contribute to the evaluation of the merit function and of a number of constraints. Finally, further details on the solution of the maximum annual energy production problem are given in §3.5. The paper is complemented by Section 4, that demonstrates the proposed procedures on the design optimization of multi-MW wind turbines (§4.1 and 4.2) and the optimal sizing of an aero-elastically scaled blade for a wind tunnel wind turbine model (§4.3).

2

Multi-disciplinary Optimization Algorithm

In this work we consider the multi-disciplinary optimization of a wind turbine as a multi-objective design problem [16] where one seeks a compromise between the maximization of the Annual Energy Production (AEP) and the weight of the machine. We implicitly assume that weight is well correlated with cost; however, we do not use cost here because reliable cost models are not available in the public domain. The problem is challenging for a number of reasons. First, the problem is subjected to a number of inequality constraints that translate various additional requirements (see below for details); many of these constraints are active at the optimal solution. Second, the evaluation of the merit function contributors and of the constraints can be computationally expensive, especially if using refined models of the machine and a large set of loading conditions. Although powerful methods are available for efficiently dealing with costly functional evaluations, for example using response surfaces or neural networks [18], these are often not well suited for highly constrained problems as the present one. We describe here a procedure that was devised for allowing the execution of a complete design loop on a standard desktop computer in a reasonably fast way, typically in one-two days. To satisfy this requirement, the multi-objective design is not formulated as a Pareto optimal problem [16], which would be the most natural choice, but rather using a combined cost defined as AEP divided by total weight. Furthermore, to minimize the number of expensive functional evaluations, we have devised a special two-stage procedure which is described next. It is clear that the use of parallel processing or more powerful computational resources might allow for different approaches than the one described here, or for the same approach to be executed faster.


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Cost: AEP Aerodynamic parameters: chord, twist, airfoils

Cost: AEP/weigh (or cost model if available) Macro parameters: rotor radius, max chord, tapering, …

Cost: Blade weight (or cost model if available) Structural parameters: thickness of shell and spar caps, width and location of shear webs Controls: model-based (selfadjusting to changing design)

Fig. 1: Overview of the multi-disciplinary optimization process. The optimization process is depicted in Fig. 1 and is formulated as follows: 1. At first, we compute an optimal solution that yields the maximum AEP with the minimum possible weight of the rotor blade (see §2.1). As shown later on, this problem leads to an iteration that alternates between a purely aerodynamic optimization of the blade twist and chord distributions for maximum AEP given a structural configuration, and a purely structural optimization for minimum blade weight given an aerodynamic design. The former problem also determines the weight of the other design-parameter-dependent components of the machine, i.e. drive-train and generator. This analysis is conducted for assigned values of some macro configuration parameters, typically the rotor radius, the maximum chord, and/or a blade taper measure (see §3.5). The design above is repeated for different values of the macro-parameters. This generates a family of different AEP optimal designs, each one achieving the least possible blade weight for that AEP. 2. Next, the family of optimal design solutions is interpolated with respect to the independent macro-parameters, and the design that achieves the best ratio of AEP and total weight is found by computing the maximum of the interpolated combined cost (see §2.2). The first step above involves essentially uncoupled aerodynamic and structural optimizations, which therefore can be performed at reasonable computational costs. The coupling between aerodynamic and structural solutions is brought about at the second step of the procedure, where AEP and total weight are combined together (notice that total weight is used in this stage, where the non-blade related weight components are estimated using weight models, so the weight optimization is not limited to the sole rotor blade). This step, however, still implies reasonable costs, since the combination is performed when optimizing with respect to the macro-parameters, which are usually just very few in number.


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In the present implementation the wind turbine tower is not optimized, but its presence is included in the aero-servo-elastic model (described later on in §3.1). The tower height is provided as an input parameter, and its structural and inertial characteristics are taken from an existing baseline configuration. This tower configuration is adjusted as the optimization of the rest of the machine progresses so that its natural frequencies are placed between the one-per-rev and two-per-rev frequencies in the whole operating range of rotor speeds, as common in most designs for avoiding resonant conditions. The design solution generated by the optimization illustrated in Fig. 1 can be fed to blade design refinement tools (cf. Fig. 2), which will be described more in detail in future publications, and that include: 1. 3D FEM structural models, using shell and/or solid elements, that are used for the verification of the results generated during the preliminary design phase using beam models, as well as for buckling and detailed stress and fatigue analysis of the blade; 2. 3D CFD fluid dynamic models, currently based on a Reynolds Averaged Navier-Stokes (RANS) finite volume formulation on Chimera grids [17], used for the study of the blade tip and root three-dimensional effects, the calculation of Coriolis-induced delayed separation, etc.

Preliminary design

Design refinement

Fully automated links

Fig. 2: Structural and aerodynamic design refinement tools.


2.1

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Stage 1: Maximum AEP for Minimum Blade Weight

The problem of finding the configuration that yields the maximum possible AEP with the minimum possible weight of the rotor blade is formulated as the following constrained optimization: Function (p∗a , p∗s , Py∗ , w∗ ) = MaxAEPMinBladeWeight(pa , ps , D) : Py∗ = max AEP(pa , ps , D) pa

(and p∗a = arg max AEP), pa

s.t.: ga (pa ) ≤ 0,

(1d)

= min Wb (ps , D)

s.t.:

(1b) (1c)

vtip ≤ vtipmax , wb∗

(1a)

ps

(and

p∗s

= arg min Wb ), ps

(1e)

gs (ps ) ≤ 0,

(1f)

ω(ps , D) ∈ [ωL , ωU ],

(1g)

E = LoadEnvelope(pa , ps , D),

(1h)

σ(ps , E, D) ≤ σadm ,

(1i)

(ps , E, D) ≤ adm ,

(1j)

d(ps , E, D) ≤ 1,

(1k)

δtipmax (ps , E, D) ≤ δtipadm .

(1l)

Here and in the following we use functions (as in Eq. (1a)) to highlight the input and output data of the various algorithms, which is useful for clarifying how the proposed procedures work. A generic function is indicated as (O) = FunctionName(I),

(2)

where I is a list of input quantities, and O a list of outputs. In problem (1), pa and ps are aerodynamic and structural, respectively, unknown parameters to be optimized, defined later in §3.2 and §3.3, while D is a list of given data: D = {Pr , Vin , Vout , R, H, AF, C, vtipmax , LDLC , . . .}.

(3)

Among the many possible elements in list D, we mention here specifically the rated power Pr , the wind speed range [Vin , Vout ] between the cut-in Vin and cut-out Vout speeds, the rotor radius R, the tower height H, the list AF = {. . . , AFi , . . .} containing the airfoil types used along the blade span, the wind turbine class C [13], the maximum allowed tip speed to limit noise emissions vtipmax [19], and the list LDLC = {. . . , DLC i.j, . . .} containing all Design Load Conditions (DLCs) [13, 14] that one wants to consider in the optimization of the machine. For a discussion on the inclusion of some of these parameters among the optimization unknowns, see later on §2.2. Problem (1) seeks a maximum for the AEP function, described in detail in §3.4. The optimal AEP is noted Py∗ , and the aerodynamic parameters describing the corresponding wind turbine configuration are noted p∗a (see Eq. (1b)). The problem is subjected to three sets of constraints. The first is given by Eqs. (1c), that in general are used to expressed desired conditions on the unknown aerodynamic parameters (often in the form of simple lower and upper bounds, for example to limit the maximum blade chord so as to satisfy transportability constraints). The second constraint ensures that the blade tip velocity does not exceed a given limit, to contain noise emissions; the effect of this constraint on the computation of the AEP is explained in §3.4. The


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third constraint is the optimal blade weight problem expressed by Eqs. (1e–1g); hence problem (1) finds the machine that maximizes the AEP subjected to the constraint, among others, of having a rotor blade of minimum weight. The optimal blade weight wb∗ minimizes the blade weight function Wb , which is computed on the detailed structural blade model of §3.3; the associated optimal structural parameters are noted p∗s (see Eq. (1e)). The total weight w of the machine is computed as the sum of the drive-train and generator weight wdt+g , plus the weight of the B blades, i.e. w = wdt+g + Bwb∗ .

(4)

It is assumed that the weight of the other components of the machine are almost constant with respect to the design parameters, at least within a certain machine category, and hence can be neglected in an optimization, since their gradients are essentially null. The weight of drive-train and generator are given by a weight model in terms of the rated power, maximum rotor speed and maximum torque, i.e. wdt+g = Wdt+g (Pr , Ωmax , Tmax ). (5) The rated power is an assigned input to the problem; the maximum rotor speed is either given by the tip speed constraint or by the rated rotor speed, and the maximum torque can be computed once the regulation policy is known (see below §3.4.2). Even the optimal blade weight problem (1e) is subjected to a number of constraints, which are detailed next. • Equations (1f) express bounds or other more complex desired conditions on the unknown structural parameters (e.g., constraints on the maximum relative position between sectional center of gravity and pitch axis, or limits on ply taper rates to account for the fact that the increase/decrease of the number of plies in a composite laminate must satisfy certain ply per length constraints imposed by manufacturing and technological considerations). • Inequality (1g) constrains some natural frequencies ω of the structure to lie within the admissible bounds [ωL , ωU ], to avoid resonant conditions in the operating envelope of the machine. For example, a typical condition is the requirement for the first blade flap natural frequency ω1 flap , which is the lowest blade eigenfrequency for a conventional configuration, to be larger than the three-per-rev frequency at the rated rotor speed ω3P (Ωr ), i.e. ω1 flap ≥ s1f ω3P (Ωr ),

(6)

where s1f is an appropriate multiplicative factor which ensures a safe gap between the two frequencies. • Equations (1h) define the load envelope E, i.e. the extreme loading conditions at all points of interest obtained by computing all DLCs in list LDLC ∈ D; the exact definition of E is deferred to §3.3. • At a number of cross sections along the blade span, the maximum stress components are noted σ(ps , E, D); the notation highlights the fact that these quantities depend on the structural configuration ps , load envelope E (which in turn depends on pa , ps and D, cf. Eq. (1h)) and given data D. Inequality (1i) constrains the maximum stresses to be lower than a given admissible upper limit σadm ; according to Refs. [13, 14], safety factors are included in the DLC loads, and therefore are automatically accounted for in the load envelope.


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• Similarly, inequality (1j) constrains the maximum strains . Details on the computation of stresses and strains are deferred to later on in this document (§3.3) for clarity of exposition. • Inequality (1k) constrains the damage caused by loads encountered in turbulent wind conditions (DLC 1.2 [13]). The definition of the load envelope used in this work includes also the load time histories in turbulent conditions, as explained in §3.3; to highlight the fact that damage depends on the structural configuration, turbulent loads contained in E and D, we use the notation d(ps , E, D). Damage at a point on a cross section due to a single stress component σr is computed as follows [13]: dσr =

X i,j,k

FVk

n(σmi , σaj , Vk ) , N (σmi , σaj , σadm , γ)

(7)

where n(σmi , σaj , Vk ) is the number of cycles computed by rain-flow counting at the mean stress σmi and amplitude σaj for wind speed Vk , N (σmi , σaj , σadm , γ) is the number of cycles to failure and γ a partial safety factor [20], while FVk is the ratio between the time spent in the life-time of the machine at wind speed Vk and the duration of the turbulent simulations. From the uni-axial damage dσr , (where r = 1 for longitudinal, r = 2 for lateral and r = 6 for shear stress components) a multi-axial damage d is computed according to Refs. [21, 22] as 2/m 1/m d = d2/m + d2/m (8) σ1 + dσ2 − (dσ1 dσ2 ) σ6 , where m is the inverse slope of the S-N curve. Damage d is computed at a number of points for each cross section of interest, and collected in a vector of damage indices d, which are then bounded to 1 in Eq. (1k). • Finally, inequality (1l) constrains the maximum blade tip deflection δtipmax measured throughout all DLC simulations in list LDLC , i.e. δtipmax = max max δtip (t, DLC). LDLC

t

(9)

The definition of the load envelope used in this work includes also the loads that cause the maximum tip deflections, as explained in §3.3; similarly to what done previously, we use the notation δtipmax (ps , E, D) to highlight this fact. Problem (1) is a nested constrained optimization problem, i.e. an optimization problem that has among its constraints a second constrained optimization problem. The direct solution of problem (1) may require a significant computational effort. A very considerable simplification of the problem may be obtained by realizing that the AEP of a machine depends to a large extent on its aerodynamic design, and considerably less so on its structural one (i.e. it strongly depends on airfoils, chord and twist distributions along the blade span, rotor radius, tower height, etc., and much less so on the thickness of the external shell of the blade, the location and sizing of the spars, etc.). Under this hypothesis, the nested problem (1) can be solved by two consecutive constrained optimizations: the first maximizes the AEP and finds the corresponding optimal aerodynamic parameters assuming frozen structural ones, while the second minimizes the blade weight and finds the optimal structural parameters by using the optimal aerodynamic ones obtained through the first optimization. If one suspects a coupling between structural parameters ps and AEP,


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the procedure can be iterated until convergence. The resulting simplified sequential constrained optimization algorithm can be expressed as follows Function (p∗a , p∗s , Py∗ , w∗ ) = SequentialMaxAEPMinBladeWeight(pa , ps , D) :

(10a)

do

(10b)

(p∗a , Py∗ ) = MaxAEP(pa , ps , D), (p∗s , w∗ ) = MinBladeWeight(p∗a , ps , D), ∆pa = kp∗a − pa k, ∆ps = kp∗s − ps k, pa = p∗a , ps = p∗s ,

(10c) (10d) (10e) (10f)

while (∆pa ≥ tolpa and ∆ps ≥ tolps ).

(10g)

The first of the two optimizations in (10) is the maximum AEP problem that, from (1), writes Function (p∗a , Py∗ ) = MaxAEP(pa , ps , D) : Py∗ = max AEP(pa , ps , D) pa

(11a)

(and p∗a = arg max AEP), pa

s.t.: ga (pa ) ≤ 0,

(11b) (11c)

vtip ≤ vtipmax .

(11d)

For further details on the solution of problem (11) see further below §3.5. The second optimization as expressed in problem (1) (see Eqs. (1e–1g)) can also by itself imply very considerable computational costs, since it requires a computation of the load envelope E for each change in the structural design variables ps . A very considerable simplification is obtained by using an iterative approach where the load envelope is considered as frozen at each step: Function (p∗s , w∗ ) = MinBladeWeight(pa , ps , D) :

(12a)

E = LoadEnvelope(pa , ps , D),

(12b)

do

(12c)

(p∗s , w∗ ) 0

= MinBladeWeightFrozenLoads(pa , ps , D, E),

E = LoadEnvelope(pa , p∗s , D), ∆ps = kp∗s − ps k, ∆E = kE 0 − ps = p∗s , E = E 0 ,

(12d) (12e)

Ek,

while (∆ps ≥ tolps and ∆E ≥ tolE ),

(12f) (12g) (12h)


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where a minimum weight blade structure for a given load envelope is computed as Function (p∗s , w∗ ) = MinWeightBladeFrozenLoads(pa , ps , D, E) : p∗s

∗

(and w = arg min W ),

= min W (ps , D)

ps

ps

s.t.:

(13a) (13b)

gs (ps ) ≤ 0,

(13c)

ω(ps , D) ∈ [ωL , ωU ],

(13d)

σ(ps , E, D) ≤ σadm ,

(13e)

(ps , E, D) ≤ adm ,

(13f)

d(ps , E, D) ≤ 1,

(13g)

δtipmax (ps , E, D) ≤ δtipadm .

(13h)

Within the minimum weight optimization at frozen load envelope, stresses, strains, fatigue damage and maximum deflections are computed for varying structural parameters, but keeping the loads fixed. The use of frozen loads is based on the hypothesis that the load envelope E changes slowly with respect to changes in the structural design parameters ps , which is a reasonable assumption in this case. Note that the effects of the design on the load envelope is just temporarily frozen to reduce the computational cost, and it is recovered by the iteration in (12). The constrained optimization problems (11) and (13) can be solved in a variety of ways. When refining an already good design solution, which hence provides an initial guess close to the optimal one, gradient-based methods can be used effectively; for the solution of both problems we have used the implementation of the sequential quadratic programming (SQP) method available in the fmincon routine of the Matlab software [23]. For the solution of problem (11), we have also used the commercial optimization software Noesis Optimus [24], which offers a choice of different global and local optimization methods and supporting functional approximation techniques that allow for a more thorough search of the design space.

2.2

Stage 2: Combined AEP and Total Weight Cost with Expansion of the Number of Design Parameters

After having computed an optimal solution using algorithm (10), it is in general useful to free one or more of the assumed parameters in list D. For example, one might be interested in optimizing the solution with respect to the rotor radius R; this might have a complex repercussion on the solution, since a different radius will not only modify the area swept by the rotor, but will also imply different loads and, for a given maximum tip speed, a different rated rotor speed and hence a different region II1/2 in the power curve, and many other effects, such as a change in the weight not only of the blade but also of the drive-train and generator. Clearly, there are several parameters other than the radius that an analyst might want to study for identifying trade-offs and evaluating design sensitivities, such as the solidity, maximum blade chord, etc. The optimization of the ratio of AEP and weight with respect to parameter Di ∈ D can be formally written as Py∗ , Di w ∗ s.t.: Di ∈ [DiL , DiU ], max

(p∗a , p∗s , Py∗ , w∗ )

= SequentialMaxAEPMinBladeWeight(pa , ps , D),

(14a) (14b) (14c)

3 Models and their Implementations

12

where DiL and DiU are lower and upper, respectively, allowable bounds on Di . Often, optimizations with one or two unknown variables as the one in problem (14), can be solved by simply computing (14c) for different values in the range [DiL , DiU ]. Interpolation of these family of optimal design points, for example using cubic splines, gives a way to readily and inexpensively compute the maximum of the merit function.

3

Models and their Implementations

The optimization procedures described in Section 2 require the ability to define parametric aeroservo-elastic models of a wind turbine, and to perform a variety of simulations on the model for each instantiation of its parameters. Such simulations include the computation of a number of given DLCs, the generation of the charts of power, torque and thrust versus tip-speed-ratio, the computation of power curves in turbulent wind, the evaluation of the natural frequencies of the structure and the associated Campbell diagram, etc. In this work, the aero-servo-elastic models are based on the multibody formulation that is briefly reviewed in §3.1. The aerodynamic parameterization is simply obtained by parameterizing the twist and chord span-wise distributions of the lifting lines of the model, as described in §3.2. For efficiency, the multibody formulation uses beam models for describing blades and tower. On the other hand, the structural parameterization used by the optimization problem is based on detailed structural models of the blade cross sections at a number of span-wise locations; from the parametric detailed sectional models, equivalent cross sectional stiffness and inertial data are generated using the approach of Ref. [25], which leads to the characterization of the sectional beam data necessary for the definition of the multibody model. The sectional models are also used for the calculation of the maximum stresses and strains experienced at each design cross section for the considered DLCs, stresses and strains that appear among the optimization constraints. The definition of the cross sectional models and their mapping into the multibody model are described in §3.3. Next, details on the implementation of the AEP and other necessary calculations are given in §3.4.

3.1

Aero-servo-elastic Wind Turbine Model

In this work, aero-servo-elastic models of wind turbines are implemented with the software Cp-Lambda (Code for Performance, Loads and Aeroelasticity by Multi-Body Dynamic Analysis), based on a finite-element multibody formulation (see Ref. [26] and references therein). The multibody approach is based on the full finite-element method, i.e. no modal-based reduction is performed on the deformable components of the structure. Cartesian coordinates are used for the description of all entities in the model, and all degrees of freedom are referred to a single inertial frame; the formulation handles arbitrarily large three-dimensional rotations. The turbine blades and tower are modeled using geometrically exact, composite-ready beams. The formulation models beams of arbitrary geometry, including curved and twisted reference lines, and accounts for axial, shear, bending, and torsional stiffness. Joints are modeled through holonomic or nonholonomic constraints, as appropriate, that are enforced by means of Lagrange multipliers using the scaled augmented Lagrangian method [27]. All joints can be equipped with internal springs, dampers, backlash, friction and power loss models, which are used among other things to account for such effects in the gear-box and drive-train. The blade pitch system response is modeled with a second order system, while the response of the generator by a first order one; both actuator models receive commanded signals by the supervision and feedback controllers.


13

Lifting lines can be associated with beam elements and their geometric description is given in terms of three-dimensional twisted curves; for generality of the implementation, these aerodynamic reference curves are distinct from the structural reference ones they are associated with. The lifting lines are based on classical two-dimensional blade element theory, and account for the aerodynamic center offset, twist, sweep, and unsteady corrections. At a number of span-wise stations along each lifting line, the aerodynamic characteristics of the aerofoil used at that location are given using look-up tables, which store for a given number of angles of attack and Reynolds numbers the values of the sectional lift, drag and moment coefficients. Lifting lines are used here to model the aerodynamic characteristics of the blades, but also of the tower and of the nacelle. An inflow element can be associated with the blade lifting lines so as to model the rotor inflow effects; the code implements the Peters-He dynamic inflow wake model [28] and a classical blade-element momentum (BEM) model based on the annular stream-tube theory with wake swirl. Tip and hub loss models are also considered. Wind is modeled as the sum of a steady state mean wind and a perturbation wind, accounting for turbulence and/or gusts. The deterministic component of the wind field implements the transients specified by IEC 61400 [13, 14], the exponential and logarithmic wind shear models, and the tower shadow effects, which include the potential flow model for a conical tower, the downwind empirical model based on Ref. [29], or an interpolation of these two models. The stochastic component of the wind field is computed according to the Mann or Kaimal turbulence models. The turbulent wind is precomputed before the beginning of the simulation for an assigned duration of time and for a user-specified two-dimensional grid of points. During the simulation, the current position of each airstation is mapped to this grid, and the current value of the wind is interpolated in space and time from the saved data. The multibody formulation used in this effort leads to a set of non-linear partial differential algebraic equations. Spatial discretization of the flexible elements of the model using the finiteelement method yields a system of differential algebraic equations in time, that are solved using an implicit integration procedure that is nonlinearly unconditionally stable [30, 31]. The implicit nature of the scheme allows for the use of large time steps and is more appropriate than explicit schemes for the typical dynamics of rotor systems. At each time step, the resulting nonlinear system of equations is solved using a quasi-Newton scheme. The time-step length is adjusted based on an error indicator. The code supports static and transient analyses, and the computation of eigenfrequencies and eigenmodes about deformed equilibrium configurations. Automated procedures support a number of standard operations, such the computation of Campbell diagrams, the automated analysis of all IEC 61400 DLCs [13, 14], the determination of trimmed periodic conditions, the generation of CP vs. tip-speed-ratio (TSR) curves, the tracing of power curves, the determination of fatigue equivalent loads using rain-flow analysis, etc. Most of these features are used for the implementation of the optimization procedures that are the subject of the present work, as detailed in §3.4.

3.2

Aerodynamic Model Parameterization

As previously explained, the aerodynamic description of the blade is given through a curved and twisted reference (lifting) line, that is associated in the model with a number of additional span-wise varying data, which include the local chord length, chord-wise position of the aerodynamic center, sweep of the airfoil with respect to the local reference line, and tabulated airfoil characteristics. Since the reference can be modeled as a three-dimensional curve, parameterized as a non-uniform rational B-spline (NURBS), one can model any generic blade shape, including


14

one with pre-bend. For the definition of the aerodynamic optimization, we use in this work parameters associated with the span-wise distributions of twist and chord, while we regard all other quantities as given and fixed. This means that in the present implementation the airfoil types and their locations along the blade span are considered as given and are not further optimized. It would be relatively straightforward to include among the free variables the span-wise locations of the airfoils, although this was not yet attempted here. Optimizing the choice of a number of airfoils from within a set of candidates can be easily done by running the required number of aerodynamic optimizations, and selecting the best performing combination. To allow for a rich description of the span-wise twist and chord distributions while keeping the number of degrees of freedom to a minimum, we use multiplicative shape functions that deform a given baseline distribution. More specifically, indicating with η ∈ [0, 1] a non-dimensional span-wise coordinate, the twist θ(η) and chord c(η) distributions are expressed as θ(η) = sθ (η)θbl (η),

(15a)

c(η) = sc (η)cbl (η).

(15b)

θbl (η) and cbl (η) are baseline given distributions (for example, describing an existing blade), while sθ (η) and sc (η) are deforming functions sθ (η) = nθ (η)θ,

(16a)

sc (η) = nc (η)c,

(16b)

where nθ (η) and nc (η) are shape functions, and θ and c are the associated nodal twist and chord parameters, respectively. Accordingly, the optimization parameters appearing in problem (10) are defined as pa = (θ T , cT )T . (17) To ensure the necessary smoothness of the twist and chord distributions θ(η) and c(η), we use splines for the shape functions nθ (η) and nc (η). Using this formulation, one can describe realistic twist and chord distributions with very few parameters. For example, the simple use of a constant shape function associated with one single unknown parameter, allows one to uniformly scale an existing blade; two parameters allow for a linear deformation of the baseline, and so on for increasing numbers of parameters. In our experience, a first analysis can be made using constants for maximum speed, and subsequently refined using cubic splines. It is clear that using assumed functions, for example splines or NURBS, for describing unknown distributions directly, i.e. without relying on the deformation of a baseline, comes with no additional conceptual difficulty, but in general requires a larger number of parameters than with the proposed multiplicative approach.

3.3

Structural Model Parameterization

The detailed structural model of the blade comprises the following elements: • A description of the external shape of the blade, which is obtained by providing the airfoil data coordinates at each span-wise location. This information, together with the curved and twisted aerodynamic reference line described above and its associated chord length data, fully defines the external blade geometry. • A description of the blade cross sectional and span-wise internal geometry. The cross sectional definition requires, at a number of locations along the span, to define the number,


15

thickness and chord-wise location of the shear webs, the chord extension and thickness of the spar caps (or single cap), and the thickness of the external blade shell (see Fig. 3 for the currently implemented section types). The cross section can be modeled using panels made of equivalent materials, or by direct modeling of the stack sequence of plies; this more detailed second option allows one to consider inter-laminar stresses in the analysis. Mesh density parameters are associated to the structural elements to support the computation of beam-like equivalent structural blade properties and the recovery of stresses and strains, as described below. The span-wise description requires defining the location along the blade span of the starting and ending points of each shear web and spar cap, and the choice between twisted webs that remain orthogonal to each local chord or straight webs that are orthogonal to the line of maximum chord (see Fig. 4 for examples of possible structural arrangements). • For each cross section, a description of the lay-up of composite laminates of all shells, webs and caps, together with the definition of all the necessary material properties. Nonstructural mass includes surface-proportional allowances for surface coating, foam core and resin take-up in core, and span-proportional allowances for adhesive on leading and trailing edges, shear web flanges, additional resin in corners of laminated components, etc.

(A) (B)

(A)

(C)

Fig. 3: Cross section types. A: three cell with shear webs and spar caps; B: three cell box type, with single inter-web capping; C: three cell, with single capping extending fore and aft of the shear webs. The structural optimization parameters ps are defined as the chord-wise location of the shear webs and, at a number of user-defined locations along the blade span, their thickness, the chord extension and thickness of the spar caps (or single cap), and the thickness of the external blade shell. At locations other than these, the same quantities are obtained either by constant or linear interpolation of the optimization parameters. To reduce the number of unknowns ps or to express manufacturing constraints, it is possible in the code to define geometric conditions among the unknowns that are then included among the constraints Eqs. (13c); this way one can, for example, constrain two shear webs to have the same thickness, or the top and lower caps of a same spar to have the same chord extension and thickness.


(A)

16

(B)

Fig. 4: Geometry of the structural blade model. A: twisted shear webs locally orthogonal to chord line, with spar caps; B: planar shear webs, with box type configuration and caps tapering out to embrace the full root circle. From the detailed structural model of the blade, two different analysis types are performed with the sectional code ANBA (Anisotropic Beam Analysis) [25]. The first is used for defining the structural and inertial characteristics of the cross sections along the blade span, that are used as inputs for the definition of the beam models in the global multibody aero-servo-elastic wind turbine model. The computation of a possibly fully populated sectional stiffness matrix, which hence accounts for all possible couplings (flap-torsion, flap-lag, extension-torsion, etc.), is performed starting from a detailed finite element mesh of the cross section using the anisotropic beam theory of Ref. [25]. The analysis also yields all other data of interest, including location of centroid and elastic center, orientation of principal axes, sectional inertias and mass. The analysis is conducted at various sections along the blade span, chosen by the user to provide for an accurate representation of the blade characteristics; the number of such sections is usually significantly larger than the number of sections where structural optimization parameters ps are defined. The sectional analysis above provides also a set of recovery relationships, that allow for the evaluation of the state of stress and strain in each section of interest given a load condition. The number and location of the sections S k where this analysis is conducted can also be selected by j the user and is typically larger than the number of sections Sdof where the structural optimization parameters are defined (cf. Fig. 5). Using the load envelope computed by scanning all DLCs of interest, the sectional mesh is searched for the points of maximum stress and strain, that are required for the writing of constraints (13e,13f). The load envelope for section S k is noted E k = (E k max , E k min ) ∈ E and it is computed as the two matrices of extreme positive and


17

j ), and sections where loads Fig. 5: Sections where structural degrees of freedom are located (Sdof are provided from the multibody aero-elastic simulations (S k ).

negative sectional stress resultants: k max Eij = skj s.t. max max ski (t, DLC),

(18a)

k min Eij = skj s.t. min min ski (t, DLC),

(18b)

LDLC LDLC

t

t

where sk (t, DLC) is the vector of stress resultants at section k at time t for the load case DLC, and si is the ith component, i = (1, 6), in sk . Accordingly, the maximum stress σ k ∈ σ is obtained as σ k = max max σ, (19) Ek

Sk

and similarly the maximum strain k ∈ is k = max max , Ek

Sk

(20)

where σ and are, respectively, the components of maximum stress and strain. In this work, the definition of the load envelope E is expanded to include the time histories of the turbulent loads due to DLC 1.2, used for enforcing the fatigue damage constraint (1k), as well as the loads associated with the maximum blade tip deflection of Eq. (9), needed for enforcing constraint (1l). This is important, because it enables the freezing of the load envelope in (12) and (13) not only for the computation of stresses and strains, but also for evaluating fatigue and maximum deflections. Otherwise, one would have to run at each change in the design parameters all turbulent simulations for varying mean speeds and all necessary DLCs for finding the maximum deflection using Eq. (9), which would be prohibitively expensive. To further reduce the cost of evaluating fatigue damage, the implementation of inequality constraint (13g) is as follows. After each new envelope evaluation, where also the turbulent loads due to DLC 1.2 are computed, the multi-axial damage index is evaluated at all points of fatigue


18

verification on all cross sections of interest. The points for which d > 1 − told (told being a positive user-defined threshold), i.e. the points where the fatigue damage constraint is either violated or close to be violated, are collected in a list. Then, for each instantiation of the design parameters, stresses are computed at these points, rain-flow counted and used for enforcing the damage constraint (13g). At the next load envelope computation, the list of fatigue damage verification points is updated. This way, one can avoid verifying the damage constraint at a large number of points, which could be potentially expensive. The computation of the loads for maximum deflection proceeds as follows. Once the list of DLCs has been searched and the maximum deflection δtipmax found (Eq. (9)), we save the values of the deflections δ j and internal stress resultants sj measured along the blade at that instant of time at j = (1, N ) span-wise locations. Then, we formulate an optimization problem that looks for an equivalent set of static loads that, applied to the blade together with the inertial and gravitational loads that correspond to the rotor speed and azimuthal position at which the maximum tip deflection occurs, yield a deflection and internal resultant distribution that match as closely as possible the saved ones. To this end, span-wise equivalent static loads l(η) are expressed in terms of shape functions nl (η) as l(η) = nl (η)l, (21) where l are nodal values. By solving a static equilibrium problem using the blade beam model of the multibody wind turbine representation, one finds, at the N locations along the blade, the deflections δˆj and internal resultants sˆj associated with this set of applied loads, which can be synthetically written as δˆj = δˆj (l),

j = (1, N ),

(22a)

sˆj = sˆj (l),

j = (1, N ).

(22b)

Finally, the optimal values l∗ of the nodal equivalent loads are found by solving the following constrained optimization problem ∗

l = arg min l

s.t.:

N X kˆ sj (l) − sj k j=1

ksj k

ˆ j − δj k kδ(l) ≤ ε, kδ j k

,

j = (1, N ),

(23a)

(23b)

where ε is a tolerance for the satisfaction of the inequality constraint. The cost function in problem (23) minimizes the difference between the internal stress resultant distribution under the equivalent static loads and the one corresponding to the maximum tip deflection, while the inequality constraints force the deflected shapes in these two conditions to match. The external loads include flap-wise and lag-wise shears, axial force and torsion, while the internal reactions include axial force, bending moments and torsion. Once the loads have been identified, they are stored in the load envelope structure in terms of their optimal nodal values and associated shape functions, i.e. {nl , l∗ } ∈ E. At each instantiation of the structural model parameters in problem (13), the loads are retrieved from E and applied to the blade beam model, together with the inertial and gravitational loads that correspond to the rotor speed and azimuthal position at which the maximum tip deflection occurs. Notice that, while the identified loads are temporarily frozen as explained above, the inertial and gravitational ones change at each instantiation of the design parameters reflecting changes in the blade mass.


19

The solution of a static equilibrium problem under this load condition yields the current estimate of the maximum blade deflection experienced by the blade throughout its operating envelope. This is a cheap operation, since it amounts to a simple, although non-linear, static solution, whose cost is compatible with it being within the optimization loop of problem (13).

3.4

Implementation of the AEP Function and DLCs for Structural Sizing

The computation of the AEP function, i.e. of the cost of problem (11), is particularly complex, and it is worth detailing. The steps that need to be performed are, in order, the computation of the power coefficient curves, the determination of the regulation policy, and, finally, the synthesis of an appropriate controller capable of regulating the machine across its entire operating spectrum. These three steps are detailed next. Notice furthermore, that the controller is also necessary for the transient simulation of all DLCs of interest for the structural sizing of the machine. 3.4.1

Computation of the CP − λ − β Curves

For each instantiation of the model parameters, it is necessary to define the regulation policies in regions II, II1/2 and III; in this work, regulation policies are computed according to the methodology detailed in Refs. [32, 33]. The regulation of the machine in the various operating regions is based on the knowledge of the power coefficient CP as a function of the TSR λ and of the blade pitch setting β, i.e. of the function CP = CP (λ, β). Such function is evaluated by computing curves of CP vs. λ for various values of β using a model of the machine. In this work, these curves are computed using one of two approaches. In the first and simplest of the two, we use the sole isolated and rigid rotor of the wind turbine, and we consider axial flow conditions and a uniform distribution of the wind over the rotor. These conditions imply a steady response of the rotor to the wind. The CP − λ − β curves obtained this way are ideal ones, and clearly lead to a slight over-prediction of the performance of the machine. This computation is faster and simpler than the one described next, and it can be done without a structural model of the wind turbine, since the rotor is considered to be rigid. Hence, we use this approach at the beginning of the design loop, or when using function (11) as a stand-alone module. However, in reality a wind turbine never operates in uniform axial conditions, at least because of the wind vertical profile, of the tower shadow effect and of the rotor up-tilt. Therefore, for constant in time wind conditions, the machine settles on a periodic orbit, the periodicity of the response being caused by the non-uniformity of the spatial distribution of the wind over the rotor disk. Furthermore, the deformation of the machine due to its flexibility, especially in high winds, can also have some effect on its power production: for example, the deflection of the blades can slightly change the rotor diameter, or the deflection of the tower can slightly modify the rotor up-tilt. To account for all the effects listed above and not accounted for when using the isolated rigid rotor, we have implemented the computation of the CP − λ − β curves using the complete flexible wind turbine multibody model. Using the complete model, which captures the principal causes for a periodic response of the machine, transient simulations could be run for given values of the wind speed, blade pitch setting and generator torque. Once the solution settles on a periodic orbit, the zeroth harmonic of all quantities of interest could be computed, including values of the power coefficient and TSR. Although this would clearly be the most accurate way of computing the CP − λ − β curves, in reality this approach is unsuitable for the present application, because one necessitates of a regulator capable of trimming the machine at each operating point of interest. While this problem


20

may be overcome for machines without tip speed constraints (pure region II-III regulation), since one may use simple PID regulators, this proves substantially more difficult for machines with tip speed constraints. In fact, in that case the presence of the transition region II1/2 necessitates of a coupled torque and pitch control, and this makes it difficult to automatically formulate simple regulators within the optimization loop for the sole problem of curve tracing. To address this issue, and still allow the computation of the CP − λ − β curves with a detailed flexible wind turbine model, we use the following procedure. Instead of conducting transient simulations, we perform static solutions for varying wind speeds, rotor speeds and blade pitch settings. During such simulations, we compute the deflected wind turbine configuration under the action of the steady aerodynamic and inertial loads due to a steady rotation of the rotor at a constant angular speed. The simulation also include all other steady loads, such as gravity, wind loads on the tower and nacelle, and blade-tower aerodynamic interference loads. In practice, such simulation amounts to a snap-shot of the wind turbine at a given rotor azimuthal position, where accelerations have been neglected except for the inertial loads caused by a constant rotor speed. Each one of these simulations is extremely fast to compute, since it amounts to a simple (although non-linear) static solution. By computing the solution at different values of the rotor azimuthal angle, one obtains a quasi-static picture of the periodic response of the rotor around a full revolution. Averaging over the rotor revolution, one obtains quasi-static estimates of the power coefficient CP . Notice that such estimates are obtained using a complete flexible model of the wind turbine that includes all principal causes for a periodic response, without necessitating of the use of a regulator. Clearly, by repeating such computations for varying values of the TSR and blade pitch, one can obtain the complete CP − λ − β curves. 3.4.2

Computation of the Regulation Policy

Once the CP − λ − β curves have been obtained, the regulation policy of the machine is computed using the approach of Refs. [32, 33], and briefly summarized next. In region II, power is maximized by operating for all wind speeds at the maximum value of the power coefficient, which is computed by solving the following optimization problem CPII = max CP (λ, β),

(24a)

λII , β II = arg max CP (λ, β),

(24b)

λ,β

λ,β

where λII and β II are, respectively, the constant (with respect to the wind speed) values of TSR and blade pitch that correspond to the maximum value of the power coefficient CPII . When the blade tip speed constraint (11d) is not active, the region II regulation policy is continued for increasing wind speeds until one reaches rated power, therefore entering region III. On the other hand, when the blade tip speed constraint (11d) is active, the machine has to be regulated by maintaining a constant rotor speed once Ωmax = vtipmax /R has been reached, which corresponds to a wind speed equal to VII 1/2 = vtipmax /λII . This defines the beginning of the transition region II1/2, whose role is to smoothly connect the region II and III policies by operating at constant rotor speed until rated power is achieved. In region II1/2 we maximize again the power coefficient by solving the following optimization problem for each wind speed V : II 1/2

∀V ∈ [VII 1/2 , Vr ] CP

∀V ∈ [VII 1/2 , Vr ] β II

(V ) = max CP λ(V ), β ,

(25a)

(V ) = arg max CP λ(V ), β .

(25b)

β

1/2

β


21

Notice that, differently from the region II case, the blade pitch varies for varying wind. The end of region II1/2 is achieved when, for sufficiently high winds, the machine achieves rated power, which signals the entrance into region III. Here the machine is regulated so as to maintain a constant power production as a function of wind speed until the reaching of the cutout wind speed, where the machine is either shut down or progressively slowed down. In region III, the power coefficient must be varied as a function of the TSR as follows CP = CPr

λ λr

3 .

(26)

¯ ¯ = vtip /R when In turn, the TSR varies as a function of the wind speed: λ = ΩR/V , where Ω max ¯ = Ωr otherwise. From the intersection of the region III schedule constraint (11d) is active and Ω given by Eq. (26) with the power coefficient curves, one can compute the corresponding values of the collective pitch settings for each TSR, and hence for each wind speed, therefore obtaining the blade pitch wind schedule β = β(V ). 3.4.3

Automatic Synthesis of a Pitch-Torque Controller in Support of the Optimization

For the evaluation of the AEP function in turbulent conditions, it is necessary to run transient simulations with the complete aero-servo-elastic model of the machine for varying values of the mean wind speed and for an assigned class of turbulence (DLC 1.1, cf. Refs. [13, 14]). Hence, one needs a pitch-torque controller capable of regulating the machine across its entire operating envelope. Furthermore, a controller is also required for conducting the simulation of all other DLCs needed by the optimization procedures. Control laws for these scopes can be designed in a variety of ways. However, for the synthesis of controllers within an optimization loop, one has to satisfy a number of specific requirements: a) for complete automation of the procedure, the controller must give reasonable performance without necessitating of manual tuning for each new instantiation of the machine; b) for simplicity, it is highly preferable to use one single controller over the entire operating envelope, which means that the controller must be capable of regulating the machine in regions II and III, but also in the special transition region II1/2. A simple controller that satisfies these conditions is the collective-pitch/torque linear quadratic regulator (LQR) described in Refs. [32, 33]. The controller is model-based, i.e. it is directly synthesized from a reduced model of the machine. Such model is readily available once a new instantiation of the machine is computed, so that the controller automatically adjusts itself to the machine design; this is a crucial advantage that allows for the automatic update of the control laws as the design of the machine evolves within the design optimization loop. Furthermore, the controller covers all operating regions of the machine, including the transitional one, in a unified manner: its multi-input-multi-output design enables a primarily constant pitch – variable torque control logic in region II, a primarily variable pitch – constant torque logic in region III, and a coupled variable pitch – variable torque logic in region II1/2 without the need of devising switching strategies in the three operational regions. The full-state-feedback LQR regulator is based on the linearization at each trim point of the operating envelope of the machine of a reduced non-linear wind turbine model. The model includes drive-train shaft dynamics, elastic tower fore-aft motion, blade pitch actuator dynamics and electrical generator dynamics. The rotor aerodynamic force and torque do not rely on a simplified aerodynamic model, but used directly the power (and force) coefficients computed using the complete aero-servo-elastic model as described above in §3.4.1. This way, the reduced


22

model inherits the aerodynamic representation of the detailed model, while keeping a very simple implementation and extremely low computational cost. The controller operates with the same logic in region II, II1/2 and III, by tracking goal regulation states (or outputs) and control inputs, tabulated in terms of the mean wind V ; all goal quantities are obtained as part of the determination of the regulation policies described in §3.4.2 (see Refs. [32, 33]). Notice that there is no switching logic in the controller when transitioning from one region to another: the control law simply tries to regulate the machine around set points for states (or outputs) and inputs, which have been computed as functions of the mean wind speed V . Since these set points vary smoothly throughout the whole operating regime of the machine, there is no need to introduce switching logics at the level of the feedback controller, therefore greatly easing its implementation and tuning. LQR gain matrices are computed at various discrete values of the mean wind, so as to cover the entire operating range of wind speeds. Similarly, goal regulation quantities are pre-computed at the same discrete wind values. During transient simulations, measurements of the instantaneous turbulent hub-height wind are either provided by the on-board anemometer or by a hub wind observer [34, 35], and in turn an estimate of the mean wind is obtained using a moving average. The estimated wind speed is used for interpolating the stored optimal gain matrices and goal regulation quantities at the current time instant. 3.4.4

Transient Simulation of DLCs

Once the regulation policies have been determined, and a controller synthesized, all DLCs of interest for the sizing of the machine can be readily simulated using the multibody wind turbine model. The Cp-Lambda code supports the fully automated analysis of all IEC 61400 DLCs [13, 14]; the data generated during the simulations is processed for extracting all relevant information required for the optimization process. In the case of the AEP, turbulent simulations according to DLC 1.1 are conducted for 600 sec for different values of the mean wind speed between the cut-in and cut-out limits, and the generated power is then weighted with the given Weibull probability density function(s) to give the annual energy production.

3.5

Solution of the AEP Maximization Problem

The solution of the AEP maximization (11) may deserve some attention in certain situations. To illustrate this fact, consider, among the possible various examples, the case of a problem with a tip speed constraint, together with an upper bound on the maximum chord to guarantee the transportability of the blade. As previously explained, the presence of the tip speed constraint, for a machine with a large rotor, creates the necessity of the transition region II1/2 in the power curve. Since the machine in that case can not be regulated up to rated power using the maximum power coefficient of region II, such situation creates a lowering of the power curve in the transition region with respect to the case without tip speed constraint; clearly, this creates a reduction in the AEP. This effect can be mitigated by increasing the solidity of the rotor. In fact, a larger solidity implies a lower rotor speed for rated power; hence, the gap between the maximum rotor speed allowed by the tip speed constraint and the rated rotor speed is reduced, which reduces the extent of region II1/2, and consequently the decrease in AEP. This means that the search for optimality of problem (11) will favor solutions with larger solidity; however, when the maximum blade chord hits its maximum allowed upper bound, larger solidities will only be generated by reducing the tapering of the blade. Although blades with

4 Applications and Results

23

little or no tapering will have lower CP ’s than blades with a more pronounced tapering, the reduction in CP is more than offset by the increase in AEP caused by the larger solidity. This means that the direct solution of problem (11) might tend to generate AEP optimal solutions that have blades with very little tapering. Such blades might incur in larger loads than tapered blades, and hence might weight more. A way to study this problem and understand its implications on a specific design, is to define a family of blades with varying tapering. The taper measure is used as one of the macro-parameters of the second stage of the optimization; this way, by solving the combined optimization, one will identify the blade with the taper measure that gives the optimal combined cost. In this work, the blade family is obtained as follows: 1. An optimal AEP blade is computed as previously explained. 2. The taper measure of that solution is computed; for example, one possible taper measure is the non-dimensional span-wise location of the blade area centroid, i.e. R1 c(η)η dη τ= 0 , (27) Sb where η ∈ [0, 1] is the non-dimensional span-wise coordinate, c the local blade chord and Sb the blade area. The taper measure for the solution obtained at step 1. above is noted τ0 . 3. The AEP maximization (11) is solved again, by adding an upper bound τ1 on the taper measure among the inequality constraints (11c): τ − τ1 ≤ 0,

(28)

where τ1 < τ0 . This will yield a solution with a lower AEP than the one associated with τ0 , but with a more pronounced tapering, and hence possibly reduced loads. 4. The previous step is repeated again, for τ2 < τ1 , and so on, until enough members of the blade family are generated. For each one of the family members, a minimum weight structural sizing is performed by solving problem (12). Each family member is identified by its value of the taper measure τi , i = (0, Nf ), and it has an associated AEP Py∗ (τi ) and an optimal weight w∗ (τi ). It is then easy to compute the optimal value of τ that corresponds to the maximum of the function Py∗ (τ )/w∗ (τ ), this way identifying the blade within the family that achieves the best compromise between energy yield and weight. This procedure is illustrated in the examples section. A part from the blade tip speed problem discussed here, more in general the constraints on the aerodynamic optimization parameters expressed by Eq. (11c) can be used for generating families of candidate blades that answer to specific requirements, while the two stage optimization approach described in Section 2 provides for a simple way of identifying within each family the blade that achieves the best possible compromise.

4 4.1

Applications and Results Design of a 2.8 MW Wind Turbine

We consider the optimization of a multi-MW machine, whose principal characteristics are given in Table 1.


24

Pr Vin , Vout R H C vtipmax cmax croot LDLC Airfoil types and span-wise locations

2.8 MW 3, 25 m/sec 49 m 84 m III A 75 m/sec 4.3 m 2.492 m 1.4, 1.5, 1.6, 1.7, 2.2 DU00-W2-401 @ η = 0.114 DU97-W-300 @ η = 0.16 DU91-W2-250 @ η = 0.25 DU93-W-210 @ η = 0.614 DU95-W-180 @ η = 0.75

Tab. 1: Principal given design characteristics of the 2.8 MW wind turbine. The aerodynamic optimization uses multiplicative degrees of freedom to deform a baseline blade configuration. The blade twist is described using 4 parameters, with degrees of freedom located at the span-wise stations η = 0, 0.2, 0.5, 0.8, while 3 parameters are used for the blade chord, with degrees of freedom located at the span-wise stations η = 0.2, 0.5, 0.8. The blade has a pre-cone of 3 deg, and no pre-bend. The blade structural layout uses a box type three cell configuration, with a single cap confined within the two shear webs (see Fig. 3(B)). The shear webs are planar, i.e. they do not follow the twist of the sections, and orthogonal to the line of maximum chord; they begin at the span-wise station η = 0.1 and end at η = 0.98. The root region of the blade uses the configuration depicted in Fig. 4(B), where the caps are gradually extended to occupy a larger chord fraction, until they brace the full root circle. The thickness of the root section is set to 80 mm, to accommodate the connecting bolts. The structural optimization parameters ps are defined as the common thickness of the two shear webs, the common thickness of the upper and lower caps, and the thickness of the external blade shell; these unknown parameters are located at the span stations η = 0, 0.02, 0.04, 0.07, 0.2, 0.4, 0.6, 0.8, and 0.98. The blade is made with two different materials, whose principal mechanical properties are given in Table 2; the unidirectional carbon fiber is used for the caps, while the biaxial glass fiber for the external shell and the shear webs. Non-structural mass is accounted for with both surface-proportional and span-proportional quantities. The minimum weight structural sizing is based on the DLCs of Table 1, which include the safety factors prescribed by Refs. [13, 14]; the structural optimization problem also includes the flap frequency placement constraint (6) with s1f = 1.2, and a maximum tip deflection constraint with δtipadm = 5 m. Loads from the multibody DLC simulations are provided at the same spanwise locations of the sectional structural parameters, where constraints (13e,13f) are enforced. A family of three AEP-optimal blades was generated using the procedure described in §3.5, and for each solution an optimal weight sizing was performed; both the aerodynamic and structural optimization steps were performed using the SQP algorithm. The principal characteristics of the computed blade family members are reported in Table 3. The effect of the taper constraint on the extent of region II1/2 is apparent, which in turn induces an effect on AEP.


25

Material Type ρ [Kg/m3 ] EL [N/mm2 ] G [N/mm2 ] σL,max [N/mm2 ] σL,min [N/mm2 ] τmax [N/mm2 ] L,max [%] L,min [%] γmax [%]

Unidirectional Carbon Fiber 1970 45900 4280 620 -373 29 1.35 -0.81 0.68

Biaxial Glass Fiber 2063 19200 3950 193 -226 53 1.0 -1.18 1.34

Tab. 2: Material properties for the 2.8 MW wind turbine. τ 0.38 0.40 0.42 0.429a a

Py∗ [MWh] 8.727e3 9.134e3 9.296e3 9.312e3

CPII

λII

0.4910 0.4911 0.4882 0.4867

10.05 9.01 8.15 7.90

VII 1/2 [m/sec] 7.4 8.4 9.2 9.5

Vr [m/sec] 12.9 12.0 11.4 11.2

wb∗ [Kg] 17331 11107 10546 10261

No constraint on τ .

Tab. 3: Principal characteristics of the blade family members for the 2.8 MW wind turbine. Figure 6 shows the span-wise chord distribution of the blade family members; the taper constraint moves the blade area in-board, while clearly also affecting the solidity since the maximum chord constraint is active for all blade solutions. It is interesting to observe that increased tapering comes with an increase in blade weight wb∗ . One might initially be tempted to think that the opposite should be true, since less tapered blades have more chord outboard along the span and hence should experience larger bending loads, which in turn should induce a weight penalty. In reality, this blade is not sized by extreme loads, and in fact the stress and strain constraints (13e,13f) are mostly inactive at convergence. The blade sizing is driven in this case by the tip deflection and frequency placement constraints, that become harder to solve for more tapered blades, and this explains the lower weights of the less tapered solutions of Table 3. Figure 7 shows the span-wise distribution of some of the computed weight-optimal structural parameters. The cap shows the classical pronounced thickening halfway along the span; the thickness span rates are compatible with typical values of ply tapering. Since the maximum blade deflection is an active constraint at convergence, it is possible that further weight saving could be obtained adding some pre-bend to the blade, although this was not attempted here. Finally, the compromise between AEP and weight was analyzed by using stage two optimization, although in reality in this particular case the solution is self evident, since increases in AEP are accompanied by decreases in weight (see Table 3). The behavior of the combined cost Py∗ /wb∗ vs. the taper parameter τ is given in Fig. 8. Clearly, the blade that achieves the best compromise is the one with a taper parameter τ = 0.429, and it is therefore assumed as the final result of the present design process. Although the problem in this case does not present a maximum, the trend of Fig. 8 can be used for predicting possible further improvements in the design obtainable


26

4.5

τ = 0.40 τ = 0.42 τ = 0.429

4 3.5

Chord [m]

3 2.5 2 1.5 1 0.5 0 0

10

20

30

40

50

Blade span [m]

Fig. 6: Chord shape of the blade family members for the 2.8 MW wind turbine. with reductions in tapering, i.e. for an increase in the τ parameter (obtainable by changing the sign in constraint (28)).

4.2

Design of a 3 MW Wind Turbine

To illustrate the complex and different effects that constraints can have on a design, we consider here the optimization of a 3 MW wind turbine, with a 53.2 m radius and a maximum chord of 3.9 m. To limit the discussion, we do not report all data used in the analysis, but just mention that is was performed in a way very similar to the one of the previous example. In particular, the aerodynamic optimization was conducted for different values of the τ parameter, and for each of the members of the family of optimal aerodynamic designs a weight-optimal structural sizing was performed. This yielded the planform shapes of Fig. 9 and the results of Table 4. τ 0.41 0.43 0.44a a

Py∗ [MWh] 1.334e4 1.354e4 1.356e4

CPII

λII

0.4930 0.4898 0.4877

9.241 8.396 8.191

VII 1/2 [m/sec] 8.2 9.0 9.2

Vr [m/sec] 11.5 10.9 10.8

wb∗ [Kg] 13993 13206 13426

No constraint on τ .

Tab. 4: Principal characteristics of the blade family members for the 3 MW wind turbine. The behavior of the combined cost Py∗ /wb∗ vs. the taper parameter τ is given in Fig. 10. It appears that in this case the merit function has a maximum around τ = 0.43. An analysis of the solution can help explain the different behavior than the one observed in the 2.8 MW case. In fact, the longer blade span and small maximum chord of the present design is penalized by excessive outboard chords, since this lowers the flap frequency and increases loads and in turn tip deflections, so that a compromise solution of intermediate taper emerges as the optimal one.


27

80 τ = 0.40 τ = 0.42 τ = 0.429

70

Thickness [mm]

60

50

40

30

20

10

0

0

5

10

15

20

25

30

35

40

45

50

Blade span [m]

80 τ = 0.40 τ = 0.42 τ = 0.429

70

Thickness [mm]

60

50

40

30

20

10

0

0

5

10

15

20

25

30

35

40

45

50

Blade span [m]

8 τ = 0.40 7

τ = 0.42 τ = 0.429

Thickness [mm]

6

5

4

3

2

1

0

0

5

10

15

20

25

30

35

40

45

50

Blade span [m]

Fig. 7: Span-wise distribution of shell (top), spar cap (middle) and web (bottom) thicknesses for the blade family members.

4.3

Design of an Aero-elastically Scaled Blade for the Wind Tunnel Model of a Wind Turbine

The procedures presented in this work can be applied to problems other than the classical sizing of a blade analyzed in the previous example. To illustrate this fact, we consider here the problem of the design of an aero-elastically scaled blade for a wind tunnel model of the V90-2.0 MW

28

AEP/wb [MWh/Kg]


0.9

0.85

0.8 0.4

0.405

0.41

0.415 τ

0.42

0.425

Fig. 8: Combined cost AEP/wb vs. taper parameter τ for the 2.8 MW wind turbine. 4

τ=0.41 τ=0.43 τ=0.44

3.5

Chord [m]

3 2.5 2 1.5 1 0.5 0 0

10

20

30

40

50

60

Blade span [m]

Fig. 9: Chord shape of the blade family members for the 3 MW wind turbine. Vestas wind turbine [36]. The model has a rotor diameter of 2 m, and hence it is termed here the V2 wind turbine. The choice of the rotor diameter was based on the size of the test section of the wind tunnel of the Politecnico di Milano (4 m by 4 m), which gives a scale factor n = RM /RP = 1/45. Here and in the following, (·)M is a quantity pertaining to the model and (·)P a quantity pertaining to the physical full scale system. The time scaling nt = tM /tP = ΩP /ΩM was set to 1/22.8, to limit the rotor speed; in fact, one of the goals of the model is the testing in closed loop of control laws, and excessive rotor speeds would imply high control bandwidths that might not be compatible with the computational resources required by advanced control laws. To have the same TSR on model and full scale machine, the wind speed scaling becomes nv = vM /vP = n/nt = 1/1.97, which implies a Reynolds number ratio ReM /ReP ≈ 1/89. The large discrepancy between full scale and model Reynolds numbers is a major concern in the design of the V2 system. To address this problem, one possible solution is the replacement of the V90 airfoils with ones specifically designed for low Reynolds applications. An investigation of the available low Reynolds airfoils lead to the selection of airfoil AH79 100C [37], used for η ∈ [0.137, 0.423], and of airfoil VM006 [37], used for η ∈ [0.654, 1], both using transition strips; in the span region η ∈ [0.423, 0.654] we used a thickness interpolation of the two airfoils, while the


29

1,03

0,99

b

AEP/w [MWh/Kg]

1,01

0,97

0.41

0.42

τ

0.43

0.44

Fig. 10: Combined cost AEP/wb vs. taper parameter τ for the 3 MW wind turbine. root region uses a cylinder, smoothly interpolated with the AH79 100C airfoil. The span-wise chord distribution was kept similar to the one of the V90 machine, while an aerodynamic optimization according to problem (11) (using CP instead of AEP as a merit function) was performed using the SQP algorithm. The effect of the optimization was to adjust the blade twist to the characteristics of the new set of airfoils, so as to achieve power and thrust coefficients as close as possible in the two machines. The resulting CP − λ − β curves are shown in Fig. 11, and compared to the ones of the V90; similarly, Fig. 12 shows the behavior of the thrust coefficient CT for varying TSR and blade pitch. In the plots, V2 quantities are shown using solid lines, while V90 ones using dashed lines; numerical values were eliminated from the axes to protect proprietary information. The power coefficient, for the same TSR and blade pitch setting, is clearly lower for the V2 than for the V90 due to the higher drag of the low Reynolds airfoils, although the overall shape of the curves is remarkably similar. The thrust coefficient on the other hand is quite similar in the two machines, due to the good lift characteristics of the V2 airfoils. The correct aero-elastic scaling of the model requires the matching of the Lock number LoM = LoP and of the lowest m non-dimensional frequencies of the system, i.e. (ωi /Ω)M = (ωi /Ω)P for i = (1, m), where m is an appropriate number that selects all modes up to a certain frequency band of interest. These requirements are satisfied if the scaled model matches, with appropriate scaling factors, the span-wise distribution of mass and stiffness of the full scale blade and tower. The structural design of the blade is particularly challenging, for its dimensions and very light weight (the scaled blade weight is about 70 grams for a span of 0.96 meters). To simplify the problem, only the lowest three blade frequencies were considered, i.e. two flap and one lag bending modes; the first torsional mode of the blade is high enough that was neglected in the design problem. In marked contrast to many aero-elastic models, the present one requires a high quality aerodynamic shape, which rules out the use of the classical non-structural disjoint segments connected to a structural carrying member. A solution better suited to the present application was found in the use of a Rohacell core, machined from a single piece to provide the shape of the blade, reinforced by two carbon fiber caps, housed in grooves obtained on the upper and lower surfaces of the Rohacell blade core; role of the carbon fiber inserts is to provide the necessary stiffness to the blade, with a minimum weight penalty. The blade surface was covered with a polymeric layer, which ensures a smooth finish and closes the cells of the Rohacell core, without


30

CP

V90 V2

TSR

Fig. 11: Power coefficient CP vs. TSR λ for different values of the blade pitch β. Solid lines: V2 scaled model; dashed lines: V90 wind turbine.

CT

V90 V2

TSR

Fig. 12: Thrust coefficient CT vs. TSR λ for different values of the blade pitch β. Solid lines: V2 scaled model; dashed lines: V90 wind turbine. substantially contributing to the sectional stiffness. This structural arrangement of the V2 blade is illustrated in Fig. 13. The chord-wise extension of the inserts and the number of plies ps was optimized at a number of control sections so as to provide the necessary local target bending stiffness, without exceeding the local target mass per unit span. This optimal constrained sizing problem can be formulated in a way similar to (13). Specifically, the optimization problem minimizes the difference between ˆ i (ps ) and the target one K i at M span-wise locations along the model bending stiffness K flap flap the blade, while matching the lowest m model natural frequencies ω ˆ j (ps ) with the corresponding j target ones ω , i.e. p∗s = arg min ps

s.t.:

M ˆ i (ps ) − K i k X kK flap flap i=1

i k kKflap

kˆ ω j (ps ) − ω j k ≤ ε, kω j k

,

j = (1, m),

where ε is a tolerance for the satisfaction of the inequality constraint.

(29a) (29b)

5 Conclusions

31

Sectional optimization variables (position, width, thickness) Span-wise shape function interpolation

Chord-wise position

Width

Carbon fiber spars for desired stiffness

Rohacell core with grooves for the housing of carbon fiber spars

Thickness

Film of glue to close pores and ensure smooth finish

Fig. 13: Structural arrangement of the V2 aero-elastic blade. The solution of problem yielded a design satisfying the specifications, since the first three blade natural frequencies were matched with errors of 0.6%, 0.06% and 0.4%, respectively. The blade was manufactured and successfully tested in the laboratories of the Politecnico di Milano [36].

5

Conclusions

In this work we have presented a comprehensive approach to the problem of optimal sizing of a wind turbine. The proposed methods were designed to support the work of an experienced analyst, by providing a complete environment where one can conduct efficiently a new sizing from scratch, a refinement of an existing configuration, or the study of trade-offs and sensitivities. The proposed procedures have the following highlights: • The coupled multi-disciplinary optimization problem has been broken down into simpler sub-problems; this not only drastically reduces the necessary computational resources, but also generates independent routines that can be profitably used as stand-alone modules (e.g., to conduct a purely aerodynamic optimization, or to conduct a structural sizing given an aerodynamic configuration). • The modeling problem uses multi-level representations of the most complicated and crucial item, i.e. the blade. Advanced non-linear geometrically exact beam models are used for supporting transient simulation needs, accounting for all coupling effects arising from the

5 Conclusions

32

use of composite materials, while sectional models based on an anisotropic beam theory are used for the local stress and strain analysis. Three-dimensional finite element models are automatically generated at the end of the optimization to support a number of design refinement and verification studies. • The procedures use a common aero-servo-elastic computational engine, which provides for high-fidelity models throughout all steps of the design process. This is crucial for correctly accounting for all effects and couplings among the different parameters and design requirements during sizing. As a further positive side effect of this approach, at the end of the optimization one automatically has a ready-to-use aero-servo-elastic model of the machine complete of associated control laws, which can be readily employed to perform whatever other analyses the user might want to conduct, for example so as to verify the solution, or to estimate loads in other conditions or in other parts of the wind turbine in order to size specific sub-components, etc. • The procedures fully automate all necessary steps that are required to conduct a complete set of analyses of a wind turbine, including the computation of the Campbell diagram, the tracing of the charts of power, torque and thrust versus tip-speed-ratio and blade pitch, the computation of the power curve, the determination of the regulation trajectories across the entire wind speed range, and the synthesis of appropriate control laws capable of good performance on all DLCs of interest. Many other less complex tasks had to be fully automated in order to implement the present design environment, including the post-processing of the results with the extraction of all relevant quantities and the automatic regeneration or update of the aero-servo-elastic and sectional models. The complete automation of all these tasks in a robust way is one of the key enabling aspects of the proposed technology, some of which involve mere software development, while others are more sophisticated and require correct technical choices, as in the case of the synthesis of control laws that should be capable of self-adjusting to changes in the design parameters. • The procedures incorporate many design requirements as constraints. We have made an effort in trying to include as many of the important constraints as possible, from frequency placements and fatigue to technological limits imposed by the use of composite materials, on the assumption that the a priori inclusion of such effects is the only way to correctly account for their mutual interactions, and that a posteriori fixes are complicated, sub-optimal and possibly expensive. Given the level of generality of the models and of the optimization algorithms, it is clear that other constraints could be easily included to account for effects that we have not considered here, or to translate specific design practices used within a company. We have demonstrated the procedures on the sizing of a multi-MW wind turbine, and the design of the blade of an aero-elastic model, with results that appear encouraging. Currently, we are applying the new set of wind turbine design tools to a number of sizing projects, which will hopefully lead to further improvements and to a rapid maturation of the technology.

Acknowledgements The contribution of M. Giuliani, C. Tibaldi and S. Dilli in the development of the software procedures described herein is gratefully acknowledged. The V2 aero-elastic wind tunnel model project is funded by Vestas Wind Systems A/S.

5 Conclusions

33

References [1] Anonymous. ECN Blade Optimization Tool BOT. ECN Wind Energy, P.O. Box 1, 1755 ZG Petten, The Netherlands, www.ecn.nl. [2] Lee K, Joo W, Kim K, Lee D, Lee K, Park J. Numerical optimization using improvement of the design space feasibility for Korean offshore horizontal axis wind turbine blade. European Wind Energy Conference & Exhibition EWEC 2007, Milan, Italy, 7–10 May (2007) [3] Maalawi KY, Badr MA. A practical approach for selecting optimum wind rotors. Renewable Energy, 28:803–822 (2003) [4] M´endez J, Greiner D. Wind blade chord and twist angle optimization using genetic algorithms. Fifth International Conference on Engineering Computational Technology, Las Palmas de Gran Canaria, Spain, 12–15 September (2006) [5] Xudong W, Shen WZ, Zhu WJ, Sørensen JN, Jin C. Blade optimization for wind turbines. European Wind Energy Conference & Exhibition EWEC 2009, Marseille, France, 16–19 March (2009) [6] Jureczko M, Pawlak M, Mezyk A. Optimization of wind turbine blades. Journal of Material Processing Technology, 167:463–471 (2005) [7] Laird D. NuMAD: blade structural analysis. 2008 Wind Turbine Blade Workshop, Sandia National Laboratories, Albuquerque, NM, USA, 12–14 May (2008) [8] Fuglsang P, Madsen HA. Optimization method for wind turbine rotors. Journal of Wind Engineering and Industrial Aerodynamics, 80:191–206 (1999) [9] Fuglsang L. Integrated design of turbine rotors. European Wind Energy Conference & Exhibition EWEC 2008, Brussels, Belgium, 31 March – 3 April (2008) [10] Anonymous. RotorOpt perfects rotor design. LM Glasfiber News Letter, September, pag. 5, (2007) [11] Duineveld NP. FOCUS5: an integrated wind turbine design tool. 2008 Wind Turbine Blade Workshop, Sandia National Laboratories, Albuquerque, NM, USA, 12–14 May (2008) [12] Jonkman J. NREL structural and aeroelastic codes. 2008 Wind Turbine Blade Workshop, Sandia National Laboratories, Albuquerque, NM, USA, 12–14 May (2008) [13] Anonymous. Wind Turbines — Part 1: Design Requirements, Ed. 3.0. International Standard IEC 61400-1 (2005) [14] Anonymous. Wind Turbines — Part 2: Design Requirements for Small Wind Turbines, Ed. 2.0. International Standard IEC 61400-2 (2006) [15] Manwell JF, Mc Gowan JG, Rogers AL. Wind Energy Explained — Theory, Design and Application. John Wiley & Sons Ltd.: Chichester, England (2002) [16] Deb K. Multi-Objective Optimization using Evolutionary Algorithms. John Wiley & Sons Ltd.: Chichester, England (2001)

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[17] Biava M. RANS Computations of Rotor/Fuselage Interactional Aerodynamics. Ph.D. thesis, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano, Italy (2007) [18] Simpson TW, Peplinski JD, Koch PN, Allen JK. Metamodels for computer-based engineering design: survey and recommendations. Engineering with Computers, 17:129–150 (2001). [19] Anonymous. Wind Turbine Generator System — Part 11: Acoustic Noise Measurement Techniques, Ed. 2.1. International Standard IEC 61400-11 (2006) [20] Anonymous. Guideline for the Certification of Wind Turbines, Ed. 2010. Germanischer Lloyd Industrial Services GmbH, Renewables Certification, Brooktorkai 18, 20457 Hamburg, Germany (2010) [21] Philippidis TP, Vassilopoulos AP. Complex stress state effect on fatigue life of GRP laminates. Part I, experimental. International Journal of Fatigue, 24:813-823 (2002) [22] Philippidis TP, Vassilopoulos AP. Complex stress state effect on fatigue life of GRP laminates. Part II, theoretical formulation. International Journal of Fatigue, 24:825-830 (2002) [23] Anonymous. Matlab. The MathWorks Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA, www.mathworks.com. [24] Anonymous. Noesis Optimus. Noesis Solutions NV, Interleuvenlaan 68, B-3001 Leuven, Belgium, www.noesissolutions.com. [25] Giavotto V, Borri M, Mantegazza P, Ghiringhelli G. Anisotropic beam theory and applications. Computers & Structures 16:403–413 (1983) [26] Bauchau OA, Bottasso CL, Nikishkov YG. Modeling rotorcraft dynamics with finite element multibody procedures. Mathematics and Computer Modeling, 33:1113–1137 (2001) [27] Bottasso CL, Bauchau OA, Cardona A. Time-step-size-independent conditioning and sensitivity to perturbations in the numerical solution of index three differential algebraic equations. SIAM Journal on Scientific Computing 29:397–414 (2007) [28] Peters DA, He CJ. Finite state induced flow models — Part II: three-dimensional rotor disk. Journal of Aircraft 32:323–33 (1995) [29] Powles SRJ. The effects of tower shadow on the dynamics of a horizontal-axis wind turbine. Wind Engineering 1983, 7:26–42. [30] Bottasso CL, Bauchau OA. On the design of energy preserving and decaying schemes for flexible, nonlinear multibody systems. Computer Methods in Applied Mechanics and Engineering, 169:61–79 (1999) [31] Bauchau OA, Bottasso CL, Trainelli L. Robust integration schemes for flexible multibody systems. Computer Methods in Applied Mechanics and Engineering, 192:395–420 (2003) [32] Bottasso CL, Croce A, Riboldi CED, Nam Y. Power curve tracking in the presence of a tip speed constraint. Renewable Energy, under review (2009) [33] Bottasso CL, Croce A. Power Curve Tracking with Tip Speed Constraint using LQR Regulators. Scientific Report DIA-SR 09-04, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, March (2009)

5 Conclusions

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[34] Bottasso CL, Croce A. Advanced Control Laws for Variable-Speed Wind Turbines and Supporting Enabling Technologies. Scientific Report DIA-SR 09-01, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano, Italy, January (2009) [35] Bottasso CL, Croce A. Cascading Kalman Observers of Structural Flexible and Wind States for Wind Turbine Control. Scientific Report DIA-SR 09-02, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano, Italy, January (2009) [36] Bottasso CL, Campagnolo F, Croce A. Development of a wind tunnel model for supporting research on aero-elasticity and control of wind turbines. 13th International Conference on Wind Engineering ICWE13, Amsterdam, The Netherlands (2011) [37] Althaus D. Profilpolaren F¨ ur Den Modellflug. Neckar–Verlag: Villingen-Schwenningen, Germany (1980)

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